# csinhf, csinh, csinhl

< c‎ | numeric‎ | complex

Complex number arithmetic
Types and the imaginary constant
 complex(C99) _Complex_I(C99) CMPLX(C11)
 imaginary(C99) _Imaginary_I(C99) I(C99)
Manipulation
 cimag(C99) creal(C99) carg(C99)
 cabs(C99) conj(C99) cproj(C99)
Power and exponential functions
 cexp(C99) clog(C99)
 cpow(C99) csqrt(C99)
Trigonometric functions
 ccos(C99) csin(C99) ctan(C99)
 cacos(C99) casin(C99) catan(C99)
Hyperbolic functions
 ccosh(C99) csinh(C99) ctanh(C99)
 cacosh(C99) casinh(C99) catanh(C99)

 Defined in header  float complex       csinhf( float complex z ); (1) (since C99) double complex      csinh( double complex z ); (2) (since C99) long double complex csinhl( long double complex z ); (3) (since C99) Defined in header  #define sinh( z ) (4) (since C99)
1-3) Computes the complex hyperbolic sine of z.
4) Type-generic macro: If z has type long double complex, csinhl is called. if z has type double complex, csinh is called, if z has type float complex, csinhf is called. If z is real or integer, then the macro invokes the corresponding real function (sinhf, sinh, sinhl). If z is imaginary, then the macro invokes the corresponding real version of the function sin, implementing the formula sinh(iy) = i sin(y), and the return type is imaginary.

## Contents

### Parameters

 z - complex argument

### Return value

If no errors occur, complex hyperbolic sine of z is returned

### Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

• csinh(conj(z)) == conj(csinh(z))
• csinh(z) == -csinh(-z)
• If z is +0+0i, the result is +0+0i
• If z is +0+∞i, the result is ±0+NaNi (the sign of the real part is unspecified) and FE_INVALID is raised
• If z is +0+NaNi, the result is ±0+NaNi
• If z is x+∞i (for any positive finite x), the result is NaN+NaNi and FE_INVALID is raised
• If z is x+NaNi (for any positive finite x), the result is NaN+NaNi and FE_INVALID may be raised
• If z is +∞+0i, the result is +∞+0i
• If z is +∞+yi (for any positive finite y), the result is +∞cis(y)
• If z is +∞+∞i, the result is ±∞+NaNi (the sign of the real part is unspecified) and FE_INVALID is raised
• If z is +∞+NaNi, the result is ±∞+NaNi (the sign of the real part is unspecified)
• If z is NaN+0i, the result is NaN+0i
• If z is NaN+yi (for any finite nonzero y), the result is NaN+NaNi and FE_INVALID may be raised
• If z is NaN+NaNi, the result is NaN+NaNi

where cis(y) is cos(y) + i sin(y)

### Notes

Mathematical definition of hyperbolic sine is sinh z =
 ez-e-z 2

Hyperbolic sine is an entire function in the complex plane and has no branch cuts. It is periodic with respect to the imaginary component, with period 2πi

### Example

#include <stdio.h>
#include <math.h>
#include <complex.h>

int main(void)
{
double complex z = csinh(1);  // behaves like real sinh along the real line
printf("sinh(1+0i) = %f%+fi (sinh(1)=%f)\n", creal(z), cimag(z), sinh(1));

double complex z2 = csinh(I); // behaves like sine along the imaginary line
printf("sinh(0+1i) = %f%+fi ( sin(1)=%f)\n", creal(z2), cimag(z2), sin(1));
}

Output:

sinh(1+0i) = 1.175201+0.000000i (sinh(1)=1.175201)
sinh(0+1i) = 0.000000+0.841471i ( sin(1)=0.841471)

### References

• C11 standard (ISO/IEC 9899:2011):
• 7.3.6.5 The csinh functions (p: 194)
• 7.25 Type-generic math <tgmath.h> (p: 373-375)
• G.6.2.5 The csinh functions (p: 541-542)
• G.7 Type-generic math <tgmath.h> (p: 545)
• C99 standard (ISO/IEC 9899:1999):
• 7.3.6.5 The csinh functions (p: 175-176)
• 7.22 Type-generic math <tgmath.h> (p: 335-337)
• G.6.2.5 The csinh functions (p: 476-477)
• G.7 Type-generic math <tgmath.h> (p: 480)

### See also

 ccoshccoshfccoshl(C99)(C99)(C99) computes the complex hyperbolic cosine (function)  ctanhctanhfctanhl(C99)(C99)(C99) computes the complex hyperbolic tangent (function)  casinhcasinhfcasinhl(C99)(C99)(C99) computes the complex arc hyperbolic sine (function)  sinhsinhfsinhl(C99)(C99) computes hyperbolic sine (sinh(x)) (function)  C++ documentation for sinh